DETERMINATION OF PARAMETERS OF HIGH TEMPERATURE INTERNAL FRICTION FROM SINGLE EXPERIMENTAL CURVE

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DETERMINATION OF PARAMETERS OF HIGH TEMPERATURE INTERNAL FRICTION FROM

SINGLE EXPERIMENTAL CURVE

Z. Jiang, X. Zhu, W. Fan

To cite this version:

Z. Jiang, X. Zhu, W. Fan. DETERMINATION OF PARAMETERS OF HIGH TEMPERATURE

INTERNAL FRICTION FROM SINGLE EXPERIMENTAL CURVE. Journal de Physique Colloques,

1987, 48 (C8), pp.C8-323-C8-327. �10.1051/jphyscol:1987847�. �jpa-00227151�

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JOURNAL DE PHYSIQUE

Colloque C8, supplement au n012, Tome 48, dkcembre 1987

DETERMINATION OF PARAMETERS OF HIGH TEMPERATURE INTERNAL FRICTION FROM SINGLE EXPERIMENTAL CURVE

Z . JIANG, X . ZHU AND W. FAN

Material Testing Center, Northeast University of Technology, Shenyang, Liaoning, China

Abstract

-

An advanced c a l c u l a t i o n has been completed by comput- e r f o r s t u d i n g A 1 and i t s a l l o y s . I n doing so, both parameters of g r a i n boundary r e l a x a t i o n and t h r e e parameters of background i n t e r n a l f r i c t i o n can be derived from s i n g l e experimental i n t e r - n a l f r i c t i o n curve containing one r e l a x a t i o n , without a c q u i r i n g any p o i n t s from experimental background curve i n advance.If two r e l a x a t i o n s p e c t r a a r e considered i n experimental curve, t h i r - t e e n parameters may be derived from s i n g l e experimental curve by computer technique. I n t h i s paper, r e l a x a t i o n s t r e n g t h i s considered a s function of temperature.

I WTRODUCTION

The high temperature i n t e r n a l f r i c t i o n of metals l i k e A 1 and i t s di- l u t e a l l o y s c o n s i s t s of g r a i n boundary and background i n t e r n a l f r i c - t i o n . I n general way, t h e background p a r t i s obtained by manual p l o t - t i n g , and t h e g r a i n boundary i n t e r n a l f r i c t i o n curve i s obtained by

s u b t r a c t i o n of background curve from o r i g i n a l experimental curve, Q ~ I v s T, where QE -1 i s experimental value of i n t e r n a l f r i c t i o n a t temera-

t u r e T. To determine t h e a c t i v a t i o n energy more than t h r e e experimen- t a l curve a r e needed. This procedure i s tedious and may l e a d t o cer-

t a i n e r r o r s . Recently Z.Y.Jiang and W. P. Cai /I/ used t h e method of computerization t o f i t experimental g r a i n boundary i n t e r n a l f r i c t i o n curve by t h e expression

/I/

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987847

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JOURNAL DE PHYSIQUE

( A / z ~ ~ ) ~ ~ ~ - ~ ' s ~ c ~ ( H ~ R ( I / T - I / T ~ ) )dw Tan9 = (2-I =

gb l+(A/J?i)~~e-wa(l+exp(2(~~~(1/~-l/~m)+w~)))-1d~ ( 1 ) where Q-I i s t h e value of g r a i n boundary i n t e r n a l f r i c t i o n , d i s t h e

relaxat% strength, v=Z/p

,

^wherez=Ln(r/ TJ,

T

i s t h e r e l a x a t i o n time, is t h e most probable value of T

,

@ i s t h e d i s t r i b u t i o n width of relax- a t i o n time 2 , H, i s t h e most probable value of g r a i n boundary a c t i v - a t i o n energy H, Tm i s t h e temperature which corresponds wtm=l.Equation

(1) shows t h a t t h e peak temperature Tpof experimental curve equals Tm only when t h e conditionp=O i s s a t i s f i e d . The above expression was de- r i v e d by A.S.Nowick and B.S.Berry /2/. By t h i s treatment, t h e authors obtained Hm,A

,

^{Tm and}

13

parameters from s i n g l e experimental g r a i n boundary curve

of

^{A 1}

-

^RE ( r a r e e a r t h element) alloys. Where

=I~,+[&/RT]

gois

d i s t r i b u t i o n width of h Ta, T~=I/,V.

,

& i s t h e frequency f a c t o r , &is t h e d i s t r i b u t i o n width of H /2/.

The authors of reference / I / used t h e following expression

t o describe t h e background i n t e r n a l f r i c t i o n f o r Al-RE alloys. Q,' i s i n s e r t e d i n t h e equation, so t h a t t h e background curve

Ln

Q" vs 1/T may be l i n e a r i n high and low temperature sides.Thereasoning was given bg i n d e t a i l i n /I/. To a b t a i n t h e constants A and B, i t i s necessary t o t a k e s e v e r a l p o i n t s ( more than s i x p o i n t s ) a t t h e p o s i t i o n f a r beyond peak a r e a of experimental curve (2;' vs T. By l i n e a r regression t o equa-

t i o n (2), parameters A and B may be determined. Then Q-l vs T background curve was deduced. Thus, t h e g r a i n boundary curve can be obtain- bg ed by s u b t r a c t i n g Q-' vs T curve from experimental i n t e r n a l f r i c t i o n curve Q;' vs T. Then using t h e computer f i t t i n g technique, f i v e para- bg meters Hm, fl ,Tm, P, andh a r e given.

According t o above method / I / , however, it i s n o t perfect. For example t h e c a l c u l a t i o n needs t o a q u i r e enough points from experimental curve a s background curve points. It i s d i f f i c u l t t o ensure whether the p o i n t belongs t o t h e background i n t e r n a l f r i c t i o n , o r g r a i n boundary i n t e r - n a l f r i c t i o n , o r both of them, e s p e c i a l l y t h e p o i n t s which a r e taken from t h e high temperature s i d e near t o t h e g r a i n boundary region. Fur- themore, r e l a x a t i o n s t r e n g t h had been regarded a s a c o n s t a n t , and i n f a c t , it has r e l a t i o n with temperature.

I n view of above f a c t s , t h i s paper introduces an advanced f i t t i n g cal- c u l a t i o n method f o r high temperature i n t e r n a l f r i c t i o n . These f a u l t s

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mentioned above w i l l be improved.

CALCULATION MOD%

If t h e d i s t r i b u t i o n of LnZobeys Gaussian one and t h e r e l a x a t i o n s t r e n g t h i s a function of temperature, i n t h e simplest case /3/,A=K/T, K i s a constant. The equation (1) can be modified a s

p

⁼

I a,

+ra/RT,

I

!The background i n t e r n a l f r i c t i o n expression i s

where may be c a l l e d a s d e v i a t i o n f a c t o r . Equation ( 3 ) and (7) can form an ob3zctive f u n c t i o n f o r c a l c u l a t i o n . when s i n g l e r e l a x a t i o n appears, t h e o b j e c t i v e function i s

M means t h e p o i n t s t h a t may be taken from experimental curve f o r fit- t i n g . If m u l t i p l e r e l a x a t i o n s appear, t h e o b j e c t i v e function i s

where jZ i s t h e v e c t o r with 5M+3 dimensions. For s i n g l e r e l a x a t i o n M = 7 , and f(?) contains e i g h t parameters. I n t h i s paper, two r e l a x a t i o n s a r e considered, thus t h i r t e e n parameters a r e calculated.

According t o equation (8) and ( g ) , t h e corresponding computer program i s worked o u t with t h e improved Powell method i n optimization / 4 / and Romberg i n t e g r a t i o n . The o b j e c t i v e function can r a p i d l y reach t o i t s minimum, and e i g h t o r t h i r t e e n high temperature i n t e r n a l f r i c t i o n parameters a r e given from s i n g l e experimental curve w i t h i n e r r o r I%.Con- sequently, s u b t r a c t i o n of background i n t e r n a l f r i c t i o n and d i v i s i o n of two r e l a x a t i o n peaks can be done by computer. With golden s e c t i o n method, t h e peak p o s i t i o n T and peak value

can

be obtained. with com-

P P

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C8-326 JOURNAL DE PHYSIQUE

mon program i n FORTRAN, computer can p r i n t o u t t h e curves Q-lvs T, Q-I v s T, Q-'vs 1/T, Q G ~ v s T and LnQ-' v s 1/T. gb

gb bg

m e r e l i a b i l i t y of equation ( 8 ) and computer program a r e checked h e r e a g a i n s t Kets e a r l y work on pure A 1 (99.99%)/5/. Hm,O, and Tp i n K e t s r e s u l t s a r e 34.5 kcal/mol, 0.5 and 285'~ r e s p e c t i v e l y . Using t h e method of t h i s paper, i.e., from s i n g l e experimental curve, t h e r e s u l t s a r e 33.0 kcal/mol, 0.51 and 2 8 8 ' ~ r e s p e c t i v e l y .

This method has been a p p l i e d i n pure A1(99.99$) and ~1-RE(RE=0.3% wt), t h e r e s u l t s a r e shown i n Appendices. The b r i e f flowchart of t h e calcu- l a t i o n and i t s computer program w i l l be published i n a n o t h e r paper, REFERENCES

/ I / J i a n g Ziying and Cai Weiping, J. De Physique, Tome46, Decembre 1985, CI O-383.

/2/ A.S .Nowick and B.S

.

Berry, Anelastic Relaxation i n C r y s t a l l i n e S o l i d s , Academic Press (1972), P92.

/3/ ^A.S. Nowick and B.S .Berry, IBM J.Res.Dev.

,

^5,297,320 ⁽¹⁹⁶¹⁾

.

/4/ D.M.Eimeblau, Applied Nonlinear programing, McGraw-Hill Book Company, N. Y. (1 972).

/5/ T.S.Ke, Phps. Rev., 71,8, (1947) P533.

High temperature i n t e r n a l f r i c t i o n of A1-RE (RE=0.3%wt) a l l a y s was measured with frequency 1Hz under t h e c o n d i t i o n i n which A1-RE a l l o y was annealed a t 4 5 0 ' ~ f o r 2hr.. Two r e l a x a t i o n s p e c t r a a r e considered and analysed by computer technique. Thirteen parameters c a l c u l a t e d by t h i s method a r e :

Note: S u b s c r i p t "1" meaas t h e low temperature peak.

S u b s c r i p t "2" means t h e high temperature peak.

The experimental high temperature i n t e r n a l f r i c t i o n curve p l o t t e d by computer i s shown i n Fig.1. The two r e l a x a t i o n peaks a r e divided by

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computer technique, and shown i n Fig.2.

High temperature i n t e r n a l f r i c t i o n of pure A 1 (99.99% w t ) was measured with frequency 1Hz under t h e c o n d i t i o n i n which i t was annealed a t 4 5 2 f o r 2hr.. Eight parameters c a l c u l a t e d by computer a r e :

The g r a i n boundary i n t e r n a l f r i c t i o n curve p l o t t e d by comouter i s shown i n Fig.3.

0.10

0.09 Fig. 1. Q;' vs T.

0.08 (1 ) O r i g i n a l experimental high

temperature i n t e r n a l f r i c t i o n curve.

0.07 (2) Grain boundary curve obtained

0.06 by computer f i t t i n g , Q-' gb v s T.

I

^0.05 ^{( 3 )}Background curve obtained by

computer f i t t i n g .

Td OM (0) Experimental p o i n t s taken from

ex e r i m e n t a l curve.

0.03

(.q

Points obtained by computer

0.02 f i t t i n g .

0.01

0 150

250

350 450

PG-

a-b

Fig. 2 d i v i d i n g two temperature peak. Fig. 3 Q;; v s T.

1. Low temperature peak.

2. High temperature peak.